GSM Phone Question

I am not familiar with the GSM system, and I didn't find an answer searching, so I'll post this question.

I live in the States right now, but will be moving to New Zealand for an extended time.

I want to buy a tri- or quad band phone to use on the T-Mobile system in the States. I was wondering if I could take the phone to NZ, have Vodafone or Telecom install a new SIM down there, so I could use the local rates while in NZ. Of course, I would want to keep the original SIM so I could go back to USA and use local rates there, as well.

TIA